Available online 19 January 2022
0950-0618/© 2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Moment-resisting beam-to-column timber connections with inclined threaded rods: Structural concept and analysis by use of the
component method
Haris Stamatopoulos
*, Kjell Arne Malo, Aivars Vilguts
Department of Structural Engineering, Norwegian University of Science and Technology (NTNU), Rich. Birkelandsvei 1A, Trondheim 7491, Norway
A R T I C L E I N F O Keywords:
Moment-resisting connection Threaded rod
Beam Column Rotational stiffness
A B S T R A C T
The use of moment-resisting frames with semi-rigid connections as a lateral load-carrying system in timber buildings can reduce the need for bracing with diagonal members or walls and allow for more open and flexible architecture. The overall performance of moment-resisting frames depends largely on the properties of their connections. Screwed-in threaded rods with wood screw thread feature high axial stiffness and capacity and they may be used as fasteners in beam-to-column, moment-resisting timber connections. In the present paper, a structural concept for a beam–to-column, moment-resisting timber connection based on threaded rods is pre- sented and explained. Analytical expressions for the estimation of the rotational stiffness and the forces in the rods were derived based on a component-method approach. The analytical predictions for stiffness were compared to experimental results from full scale tests and the agreement was good.
1. Introduction 1.1. Background
WOODSOL is a Norwegian research project which aims to develop a structural system for multi-storey timber buildings based on moment- resisting frames (abbr. MRFs) [1]. The design of multi-storey timber buildings is often governed by the fulfilment of serviceability re- quirements, namely the restriction of wind-induced accelerations and deflections and human-induced vibrations within acceptable limits.
Considering structures subjected to horizontal loading (e.g. wind), MRFs with semi-rigid, beam-to-column, moment-resisting connections as a lateral load-carrying system, can reduce the need for bracing with di- agonal members or walls and therefore allow for more open and flexible architecture. Recent studies [2,3] have shown that a minimum rota- tional stiffness of connections of the order of 10000–15000 kNm/rad is required in multi-storey MRFs, in order to fulfil the serviceability re- quirements due to wind-induced deflections and accelerations. With respect to human-induced vibrations, rotationally stiff connections at the ends of beams and floors may significantly improve their perfor- mance [3,4] allowing for longer spans. Moreover, MRFs are statically indeterminate structures and the distribution of internal forces and
moments at the Ultimate Limit State, depend on the rotational stiffness of their connections. Therefore, the overall performance of MRFs de- pends largely on their connections and an accurate estimation of the rotational stiffness is necessary in the analysis of MRFs.
Compared to axially loaded fasteners, laterally loaded fasteners feature lower stiffness. Consequently, in order to achieve the required rotational stiffness in connections with laterally loaded fasteners, a large number of fasteners and shear planes may be required. In the literature, analytical models and experimental results for moment-resisting con- nections with axially loaded fasteners can be found, mainly for two types of fasteners: inclined self-tapping screws, see e.g. [5–7] and glued-in rods inserted parallel or perpendicular to grain, see e.g. [8–10].
Axially loaded threaded rods (i.e. screwed-in rods with wood-screw threads) can be a promising alternative for such connections as they feature high axial capacity and stiffness [11,12]. To achieve a fast and economic assembly, it is better to pre-install threaded rods and coupling parts in the beam and the column and only do mounting of the coupling parts at the building site. The mounting between the parts at the building site, should be simple, reliable and should not influence the stiffness of the connection.
* Corresponding author.
E-mail address: [email protected] (H. Stamatopoulos).
Contents lists available at ScienceDirect
Construction and Building Materials
journal homepage: www.elsevier.com/locate/conbuildmat
https://doi.org/10.1016/j.conbuildmat.2022.126481
Received 3 November 2021; Received in revised form 12 January 2022; Accepted 13 January 2022
1.2. Outline
A structural concept for a moment-resisting, beam-to-column, timber connection is presented and explained in this paper. The concept is based on threaded rods which are mainly axially loaded, to take advantage of their high axial stiffness and capacity. Focus is given on the derivation and formulation of analytical expressions for the rotational stiffness and the forces in the rods since these are important inputs in the analysis and the design process. The analytical predictions are compared to experimental results from full-scale tests of prototype, moment- resisting connections between glued-laminated timber (abbr. glulam) members.
2. Conceptual design of connection 2.1. General remarks
The structural concept for a moment-resisting, beam-to-column timber connection with inclined threaded rods is presented in Fig. 1. The rods are inserted in pre-drilled holes in the beam and the column and jointed by use of coupling parts. In the study [13] which is used here to illustrate the concept, purpose-made steel rings were used as the coupling parts, see Fig. 1(d). To allow fastening of rods to the rings, threaded rods with metric thread at their end are used, confer Fig. 1(c).
A great challenge in beam-to-column moment-resisting connections is the transfer of forces between two members whose grain orientation differs by 90 degrees. Wood is 15–30 times stiffer along the grain compared to transversal directions. To utilize the higher material stiff- ness of wood parallel to grain, loading perpendicular to grain should be minimized. Moreover, threaded rods are optimized for axial loading and therefore lateral loading of the rods should be minimized to fully utilize their potential. Axially loaded rods are very stiff fasteners especially when they are installed with small inclination to the grain direction and they allow for immediate load take-up, without initial slip [12].
2.2. Column-side connection
The coupling parts are connected to the column by use of a pair of inclined threaded rods in each side (rods c1-c2 at the top and rods c3-c4 at the bottom), see Fig. 1(a). Due to the inclination of the rods and the
existence of shear forces, a load situation consisting of both axial and lateral forces occurs in the rods. However, the rods will mainly experi- ence axial forces since their axial stiffness is much greater than the lateral one. The transfer of forces in this configuration resembles the transfer of forces in a truss, where members are predominantly axially loaded. Therefore, the lateral forces in the threaded rods c1, c2, c3 and c4 may be neglected. Wood is very soft perpendicular to grain and transfer of forces by contact between the column and the coupling part at the compressive side would result in low contribution to the overall stiff- ness. Moreover, transfer of forces by contact may be influenced by po- tential shrinkage of the timber members. Therefore, transfer of forces by contact is neglected and the compressive forces are assumed to be transferred only by the threaded rods.
2.3. Beam-side connection
Threaded rods oriented parallel to the grain are vulnerable to cracks since a single crack along the grain might lead to a considerable loss of strength if the crack occurs in the same plane as the rod. Therefore, the beam is connected to the coupling parts by use of threaded rods (b1 and b2) inserted at a small angle to the grain, i.e. 5–10 deg, see Fig. 1(a).
Greater angle is avoided, as it would also result in high lateral forces in the threaded rods and therefore smaller stiffness. Moreover, a small inclination of the rods on the beam side allows for increased penetration length and higher axial stiffness of the rods. Installing the threaded rods with a small inclination, allows also to install them with very small (even zero) edge distance, resulting in increased lever arm and therefore in increased moment resistance and rotational stiffness, see Fig. 1(a and b).
This is an advantage of inclined rods compared to either rods inserted parallel to grain or laterally loaded fasteners.
2.4. Metallic coupling parts
The inclination of rods gives some practical implications in jointing the different parts. The use of steel rings as coupling parts gives the opportunity to insert threaded rods in different angles to the grain and join them in one point; in this way eccentricities are avoided. Purpose- made washers are used to fasten the threaded rods in the rings with nuts, see Fig. 1(d). To allow the assembly of the connection, the rings are sliced in the symmetry line resulting in two parts, wrapped around the Fig. 1. Moment-resisting, beam-to-column timber connection with threaded rods: (a) lay-out of structural concept, (b) section view, (c) threaded rods with wood screw thread and metric thread at their end, (d) details of the fastening of the threaded rods to the steel ring couplers.
rods and fastened together with bolts, see Fig. 1(d). Beam-to-column connections are typically subjected to alternating moment due to wind or earthquake loading. To transfer forces both in tension and compres- sion due to alternating loading, special washers and nuts are placed both in the interior and the exterior surface of the coupling rings, see the detail in Fig. 1(d).
3. Component method
3.1. Spring components and force distribution
Fig. 2(a) shows the acting forces on each part of the connection, due
to moment-loading as shown in Fig. 1(a). The analysis presented in this Section is for each plane of rods, i.e. rods inserted at the same plane.
Each rod is represented by two linear-elastic springs taking into account their axial stiffness Kax,i and their lateral stiffness Kv,i. No contact is assumed between the beam and the column or between steel rings and timber members. The distance between the centroids of the steel rings is denoted z, see Fig. 2(a). Note that in the concept in Fig. 1 the rods are connected concentrically in the rings and thus the distance z is the same for all parts of the connection, i.e.z= zbeam= zcolumn = zcon. The following abbreviations are used for the sine and the cosine values of the angles of the rods to the grain direction in the column (αci) and in the beam (αbi):
Fig. 2. Component method: (a) Acting forces on each part of the connection, (b) Finite Element model of the connection, (c)-(e) Components, forces and dis- placements at the column side (c), at the beam side (d) and in the steel coupling parts (e) (dashed lines: initial position of springs).
sci=sin(αci);cci=cos(αci);sbi=sin(αbi);cbi=cos(αbi); (1) In FE models, the beams and the columns are typically modelled as linear finite elements (located at the centroids of the elements). The connection can be modelled by a rotational spring. To take into account the fact that the beam is not continuous, it is convenient to model the rotational spring located at the edge of the column, with an offset to the centroid of the column, as shown in Fig. 2(b). Based on this modelling approach, moment is determined at the edge of the column in the pre- sent paper.
The magnitude of the horizontal force Fx per plane of rods is obtained by equilibrium (see Fig. 2(a)):
Fx=1 n∙M
z (2)
As an approximation, it is assumed that the shear force is equally distributed between the top and the bottom edge, i.e.:
Fy≈1 n∙V
2=1 n∙ M
2∙Lv (3)
where Lv is the lever arm (see Fig. 1(a)) and n is the number of planes of rods, see also Fig. 1(b).
3.2. Stiffness and forces in each rod 3.2.1. Column-side connection
The components, the forces, and the displacements of the connection between the steel coupling parts and the column are shown in Fig. 2(c).
The axial stiffness Kax,ci of threaded rods is much greater than their lateral stiffness Kv,ci, i.e. Kax,ci≫Kv,ci, and since two rods are used to connect each coupling part to the column (rods c1 and c2 at the top and rods c3 and c4 at the bottom), forces are mainly transferred in the axial direction of the rods. Therefore, the lateral springs may be neglected in this case. In fact, the following analysis considering lateral springs leads to similar results but much more cumbersome expressions.
Eqs. (4) and (5) provide the forces-displacements relation in the global coordinate system at the tensile side (rods c1-c2) and the compressive side (rods c3-c4), according to the component model in Fig. 2(c). As indicated by Eqs. (4) and (5), linear elasticity is assumed in the rods.
{Fx
Fy
}(c1−c2)
=Qc,12T∙
[Kax,c1 0
0 Kax,c2
]
∙Qc,12∙ {δx
δy
}(c1−c2)
(4) {Fx
Fy
}(c3−c4)
=Qc,34T∙
[Kax,c3 0
0 Kax,c4
]
∙Qc,34∙ {δx
δy
}(c3−c4)
(5) Qc,12 and Qc,34 are the transformation matrices given by Eq. (6):
Qc,12=
[sc1 cc1
sc2 − cc2
]
;Qc,34=
[sc3 cc3
sc4 − cc4
]
(6) Solving Eqs. (4) and (5) for the displacements in the global coordi- nate system results in Eqs. (7) and (8):
{δx
δy
}(c1−c2)
=
[ Sxx,c − Sxy,c
− Sxy,c Syy,c
](c1−c2)
∙ {Fx
Fy
}(c1−c2)
(7) {δx
δy
}(c3−c4)
=
[ Sxx,c − Sxy,c
− Sxy,c Syy,c
](c3−c4)
∙ {Fx
Fy
}(c3−c4)
(8) The compliance terms in Eqs. (7) and (8) are given by the following expressions:
Sxx,c(c1−c2)=
cc12 Kax,c2+Kcc22
ax,c1
(cc1∙sc2+cc2∙sc1)2;Sxx,c(c3−c4)=
cc32 Kax,c4+Kcc42
ax,c3
(cc3∙sc4+cc4∙sc3)2 (9)
Syy,c(c1−c2)=
sc12 Kax,c2+Ksc22
ax,c1
(cc1∙sc2+cc2∙sc1)2;Syy,c(c3−c4)=
sc32 Kax,c4+Ksc42
ax,c3
(cc3∙sc4+cc4∙sc3)2 (10)
Sxy,c(c1−c2)=
cc1∙sc1 Kax,c2− cKc2∙sc2
ax,c1
(cc1∙sc2+cc2∙sc1)2;Sxy,c(c3−c4)=
cc3∙sc3 Kax,c4− cKc4∙sc4
ax,c3
(cc3∙sc4+cc4∙sc3)2 (11) The forces at the tensile side (rods c1-c2) and the compressive side of the connection (rods c3-c4) per plane of rods are given by Eqs. (12) and (13), see also Eqs. (2) and (3) and Fig. 2(a):
{Fx
Fy
}(c1−c2)
=1 n∙
{M/z V/2 }
=1 n∙
{ M/z M/(2∙Lv)
}
(12) {Fx
Fy
}(c3−c4)
=1 n∙
{− M/z V/2
}
=1 n∙
{ − M/z M/(2∙Lv)
}
(13) The horizontal displacements are obtained by substituting Eqs. (12) and (13) into Eqs. (7) and (8):
δx(c1−c2)=1 n∙
(
Sxx,c(c1−c2)∙M
z− Sxy,c(c1−c2)∙ M 2∙Lv
)
(14)
δx(c3−c4)=1 n∙
(
− Sxx,c(c3−c4)∙M
z− Sxy,c(c3−c4)∙ M 2∙Lv
)
(15) The rotation is given as function of the horizontal displacements by Еq. (16), confer also Fig. 2(c):
θc=δx(c1−c2)− δx(c3−c4)
z (16)
Combining Eqs. (14) to (16), the rotational stiffness per plane of rods is obtained:
Kθ,c=1 n∙M
θc
= z2
(Sxx,c(c1−c2)+Sxx,c(c3−c4)) +(
Sxy,c(c3−c4)− Sxy,c(c1−c2))
∙2∙Lzv (17) Finally, the axial forces in the rods are given by Eqs. (18) and (19):
{Fax,c1
Fax,c2
}
=( Qc,12T)−1
∙ {Fx
Fy
}(c1−c2)
=1 n∙
⎧
⎪⎪
⎪⎨
⎪⎪
⎪⎩
cc2+sc2∙z/(2∙Lv) cc1∙sc2+cc2∙ sc1
cc1− sc1∙z/(2∙Lv) cc1∙sc2+cc2∙ sc1
⎫
⎪⎪
⎪⎬
⎪⎪
⎪⎭
∙M
z (18)
3.2.2. Beam-side connection
The components, the forces, and the displacements in the connection between the coupling parts and the beam are shown in Fig. 2(d). The direction of the resultant force does not coincide with the axis of the rod and here it is necessary to consider both the axial (Kax,bi) and the lateral stiffness of the rods (Kv,bi). Eqs. (20) and (21) provide the forces- displacements relations in global coordinates at the tensile (rod b1) and the compressive side (rod b2). As indicated by Eqs. (20) and (21), linear and un-coupled elasticity of the rods is assumed.
{Fx
Fy
}(b1)
=Qb1T∙
[Kax,b1 0 0 Kv,b1
]
∙Qb1∙ {δx
δy
}(b1)
(20) {Fx
Fy
}(b2)
=Qb2T∙
[Kax,b2 0 0 Kv,b2
]
∙Qb2∙ {δx
δy
}(b2)
(21)
Qb1 and Qb2 are the transformation matrices given by Eq. (22):
Qb1=
[ cb1 sb1
− sb1 cb1
]
;Qb2=
[cb2 − sb2
sb2 cb2
]
(22) Solving Eqs. (20) and (21) for the displacements in the global coor- dinate system results in Eqs. (23) and (24):
{δx
δy
}(b1)
=
[ Sxx,b1 − Sxy,b1
− Sxy,b1 Syy,b1
]
∙ {Fx
Fy
}(b1)
(23) {δx
δy
}(b2)
=
[ Sxx,b2 − Sxy,b2
− Sxy,b2 Sxx,b2
]
∙ {Fx
Fy
}(b2)
(24) The compliance terms in Eqs. (23) and (24) are given by the following expressions:
Sxx,b1= sb12
Kv,b1
+ cb12
Kax,b1
;Sxx,b2= sb22
Kv,b2
+ cb22
Kax,b2 (25)
Syy,b1= cb12
Kv,b1
+ sb12
Kax,b1
;Syy,b2=cb22
Kv,b2
+ sb22
Kax,b2 (26)
Sxy,b1=sb1∙cb1∙ ( 1
Kv,b1
− 1 Kax,b1
)
; Sxy,b2=sb2∙cb2∙ ( 1
Kax,b2
− 1 Kv,b2
)
(27) The forces at the tensile side (rod b1) and the compressive side (rod b2) per plane of rods are given by Eqs. (28) and (29), see also Eqs. (2) and (3) and Fig. 2(a):
{Fx
Fy
}(b1)
=1 n∙
{M/z V/2 }
=1 n∙
{ M/z M/(2∙Lv)
}
(28) {Fx
Fy
}(b2)
=1 n∙
{− M/z V/2
}
=1 n∙
{ − M/z M/(2∙Lv)
}
(29) The horizontal displacements are obtained by substituting Eqs. (28) and (29) into Eqs. (23) and (24):
δx(b1)=1 n∙
( Sxx,b1∙M
z− Sxy,b1∙ M 2∙Lv
)
(30)
δx(b2)= − 1 n∙
( Sxx,b2∙M
z+Sxy,b2∙ M 2∙Lv
)
(31)
The rotation is given as function of the horizontal displacements, confer also Fig. 2(d):
θb=δx(b1)− δx(b2)
z (32)
Combining Eqs. (30) to (32), the rotational stiffness per plane of rods is obtained:
Kθ,b=1 n∙M
θb
= z2
(Sxx,b1+Sxx,b2
)+(
Sxy,b2− Sxy,b1
)∙2∙Lzv (33)
Finally, the forces in the rods are given by Eqs. (34) and (35):
{Fax,b1
Fv,b1
}
=( Qb1T)−1
∙ {Fx
Fy
}(b1)
=1 n∙
{
cb1+sb1∙z/(2∙Lv) − sb1+cb1∙z/(2∙Lv) }
∙M
z (34)
{Fax,b2
Fv,b2
}
=( Qb2T)−1
∙ {Fx
Fy
}(b2)
= − 1 n∙
{
cb2+sb2∙z/(2∙Lv) − sb2+cb2∙z/(2∙Lv) }
∙M
z (35) 3.2.3. Steel connectors
The steel rings are represented by spring components with spring constants Kax,con,1 and Kax,con,2, see also Fig. 2(e). Assuming linear elas- ticity, the rotational stiffness of the connectors is given by:
Kθ,con=1 n∙M
θcon
=z2∙ ( 1
Kax,con,1
+ 1 Kax,con,2
)−1
(36)
3.2.4. Rotational stiffness of entire connection
The total deformation of the connection is obtained by adding the deformation in each part. This is equivalent to a system of rotational springs placed in series. Taking all sources of deformation into account, the rotational stiffness of the entire connection per plane of rods is given by Eq. (37):
Kθ,tot= ( 1
Kθ,c
+ 1 Kθ,b
+ 1 Kθ,con
)−1
(37)
3.3. Resistance considerations
3.3.1. Capacity of threaded rods and coupling parts
A power criterion is often used – as an approximation – to determine the capacity of fasteners subjected to combined axial force (Fax) and lateral force (Fv), i.e.:
(Fax
Fax,R
)q +
(Fv
Fv,R
)q
≤1 (38)
In Eq. (38), Fax,R and Fv,R are the axial and lateral capacity of a fastener respectively. According to EN 1995-1-1 [14], a quadratic failure criterion applies for screws, i.e. q =2. The quadratic failure criterion has provided safe-sided predictions for long self-tapping screws (i.e.
with steel failure being more critical than withdrawal) inserted perpendicular to grain [15] and for glued-in rods parallel to grain [16].
{Fax,c3
Fax,c4
}
=( Qc,34T)−1
∙ {Fx
Fy
}(c3−c4)
= − 1 n∙
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
cc4− sc4∙z/(2∙Lv) cc3∙sc4+cc4∙ sc3
cc3+sc3∙z/(2∙Lv) cc3∙sc4+cc4∙ sc3
⎫⎪
⎪⎪
⎬
⎪⎪
⎪⎭
∙M
z (19)
However, more experimental verification is required with respect to such failure criteria. The threaded rods in the column are mainly axially loaded (i.e. Fv,ci≈0) as explained in Section 2.2 and therefore in this case Eq. (38) reduces to:
⃒⃒Fax,ci
⃒⃒≤Fax,R (39)
Finally, the steel coupling parts should have sufficient resistance.
3.3.2. Failure in the panel zone of the column
The application of horizontal forces results in high shear stresses in the panel zone of the column, i.e. the region between threaded rods c1-c2
and c3-c4. Moreover, stresses perpendicular to grain occur around the threaded rods. The combination of tensile stresses perpendicular to grain and shear stresses is unfavourable due to their high degree of interaction [17] and may cause fracture in the panel zone.
3.4. Properties of threaded rods 3.4.1. Axially loaded threaded rods
The properties of threaded rods are necessary inputs for the pre- sented component method. Considering the withdrawal stiffness of a threaded rod under service load (Kser,ax) and the stiffness of the free non- embedded part (Kax,l0), the total axial stiffness of a threaded rod is given by:
Kax= Kser,ax∙Kax,l0
Kser,ax+Kax,l0 (40)
Eq. (41) is an approximation for the withdrawal stiffness of a threaded rod [18] (Kser,ax in N/mm):
Kser,ax≈50000∙(d/20)2∙(ρm/470)2∙min [
(l/300)0.75,1.0 ]
0.40∙cos2.3α+sin2.3α (41) where d is the outer-thread diameter of the rod (in mm), l is the pene- tration length of the rod (in mm), ρm is the mean density of timber (in kg/m3) and α is the angle between the rod and the grain direction. The axial stiffness of the non-embedded part is given by:
Kax,l0=Anet∙Es/l0 (42)
where:
• Es=210000 N/mm2 is the modulus of elasticity of steel
• Anet≈π∙dnet2/4 is the net area of the rod (for rods with a metric- threaded end as shown in Fig. 1(c), dnet may be approximated as 90% of the diameter of the metric thread [19]);
• l0 is the non-embedded length of the rod, i.e. the length between the fixing point of the rod in the coupling part and the entrance point of the thread in the timber, see also Fig. 2(a).
The axial capacity per threaded rod is given by Eq. (43):
Fig. 3. Full-scale tests by Lied and Nordal [13]: (a) experimental set-up, (b) simplified static system and estimated shear forces in the column, (c) lay-out of specimens and (d) failure modes.
Fax,R=nef
n∙min {Fax,α,R
Ftens,R (43)
where:
•Fax,α,R is the withdrawal capacity per rod;
•Ftens,R is the design tensile capacity of each rod;
•n is the number of threaded rods acting together (i.e. the plane of rods);
•nef is the effective number of rods acting together and according to EN1995-1-1 [14] it may be estimated as function of the number of planes of rods as:
nef=n0.9 (44)
In [18], the following simplified expression for the mean withdrawal capacity is provided:
Fax,α,Rm≈15.0∙d∙l∙(ρm/470) (45)
On principle, the buckling resistance should also be verified for rods subjected to compressive forces.
3.4.2. Laterally loaded threaded rods
The lateral stiffness of the threaded rods is an input parameter for the determination of the properties of the beam-side connection as discussed in Section 3.2.2. Assuming that rotation of the rods is not allowed at the fixing points in the coupling parts (since rods are fastened by nuts on both sides of the rings), the total vertical stiffness (Kv) of a threaded rod can be estimated by Eq. (46) [18]:
Kv=3∙kv∙lch
λ03+3∙λ02+3∙λ0+3 (46)
The parameters λ0 and lch have been defined as follows [18]:
λ0=l0/lch;lch= ̅̅̅̅̅
44∙
√ Es∙Is/kv (47)
The parameter Is≈π∙d14/64 is the 2nd moment of area of the rod and d1 is the core diameter. The parameter kv is the foundation modulus (i.e.
stiffness per unit length) of a laterally loaded rod.
The load-carrying capacity of a long, laterally-loaded rod, loaded with eccentricity l0 is given by Eq. (48) [20]:
Fv,R=fh∙def∙
( ̅̅̅̅̅̅̅̅̅̅̅̅̅̅
2∙My,R
fh∙def
√
+e02− e0
)
(48) where fh is the embedment strength, My,R is the yielding moment and def
is an effective diameter taking into account the presence of the thread.
According to EN 1995-1-1 [14] the effective diameter is approximated as 1.1 times the core diameter, i.e. def=1.1∙d1.The value of e0 depends on whether the rotation at the loading point is restrained or not: if rotation is allowed e0=l0 and if rotation is restrained e0= (l0− lch)/2 [18]. The latter is assumed to be more realistic because rods are fastened by nuts on both sides of the rings. The threaded rods are inserted with small inclination to the grain in the beam and therefore the foundation modulus kv and the embedment strength fh perpendicular to the grain may be used as approximations.
4. Experimental validation
In this Section, the analytical predictions according to the compo- nent method are compared to test results obtained by two experimental Table 1
Parameters for tests in accordance with Fig. 3 [13]
Test Parameters S35-
55–10 αc1=αc4=35◦, αc2=αc3=55◦, αb1=αb2=10◦lc1=lc4=785mm, lc2=lc3=240mm, lb1=lb2=1100mml0,c1 a =120 mm , l0,c2 a =80 mm, l0,c3 a =35 mm, l0,c4 a =75 mm, l0,b1 a =80 mm, l0,b2 a =35 mmz=
450mm, Lv=2000 mm S55-
70–10 αc1=αc4=55◦, αc2=αc3=70◦, αb1=αb2=10◦lc1=lc4=540mm, lc2=lc3=450mm, lb1=lb2=1100mml0,c1 a =80 mm, l0,c2 a =95 mm, l0,c3 a =50 mm, l0,c4 a =35 mm, l0,b1 a =80 mm, l0,b2 a =35 mmz=
450mm, Lv=2000 mm
a l0,c1∕=l0,c4, l0,c2∕=l0,c3 and l0,b1∕=l0,b2 despite geometric symmetry: the tensile loads in the top edge are transferred by contact between fixing nuts and the interior surface of the rings, while the compressive loads in the bottom edge are transferred by contact between fixing nuts and the exterior surface of the rings.
Fig. 4. FE simulations of steel rings: (a) Ring 35–55 in tension, (b) Ring 35–55 in compression, (c) Force-displacement curves for all cases.
series. In Section 4.1, test results from two prototype full-scale tests of moment-resisting, beam-to-column connections with steel rings as coupling parts [13] (as shown in Fig. 1(a)) are presented in detail and compared to the analytical predictions. In Section 4.2, test results from a similar prototype connection with a steel connector consisting of an IPE
profile and welded steel plates [21] are briefly presented and compared to the analytical predictions.
4.1. Connection with steel rings as coupling parts
The set-up for these tests [13] and the specimens are presented in Fig. 3. The specimens were tested according to EN 26891:1991 [22]. The beam and the column were made of Norway spruce glulam (Picea Abies) of strength class GL30c [23] with lamination thickness of 45 mm. The cross–sectional dimensions of both the beam and the column were b× h=140×450 mm. The moisture content was approximately 12%. The beam and the column were assembled with a gap at their interface and therefore transfer of forces by contact was not allowed, at least for small values of moment. Purpose-made threaded rods with outer-thread diameter d=22 mm, core diameter d1= 16.1 mm and metric M20 thread in one of their ends were used, see Fig. 1(c).
Two planes of rods were used (i.e. n=2). The spacing and edge distances of rods, as specified in Fig. 1(b), were a2=60 mm and a4=40 mm. The tests are named Sac1-ac2-ab1, based on the angles of the rods. The angles αi, the embedment lengths li,and the free lengths of the rods l0,i of the rods for each test are given in Table 1.
Steel rings (Fig. 1(d)) of steel quality S355 with an inner diameter of 115.2 mm and an outer diameter of 172.3 mm were used. Holes with diameter 22 mm were drilled to allow the mounting of threaded rods at their metric-threaded end, i.e. a small tolerance of 2 mm was provided to allow for easy assembly. Afterwards, the part was sliced in the symmetry Table 2
Material properties of S355 steel in FE model.
Material property Values
Modulus of elasticity Es=210000 N/mm2
Poisson’s ratio ν =0.30
Plastic strain levels at selected stress levels[25] σ=311.0N/
mm2
εp=0 σ=346.9N/
mm2
εp=0.4%
σ=355.9N/
mm2 εp=1.97%
σ=541.6N/
mm2 εp=13.91%
Table 3
Tests with rings as coupling parts (Fig. 1 and Fig. 3) – Stiffness predictions vs experimental results.
Parameter Units Reference S35-55–10 S55-70–10
Kax,c1 (kN/
mm) Eq. (40) in
parenthesis: 78.5 (95.3,
445.3) 61.8 (68.1, 668.0)
Kax,c2 53.0 (57.6,
668.0) 51.1 (56.2, 562.5)
Kax,c3 55.5 (57.6,
1526.8) 53.4 (56.2, 1068.8) Kax,c4 Kser,ax,Kax,l0 Eq.
(41),Eq.(42) 84.0 (95.3,
712.5) 65.2 (68.1, 1526.8)
Kax,b1 105.5
(125.4, 668.0)
105.5 (125.4, 668.0)
Kax,b2 115.8
(125.4, 1526.8)
115.8 (125.4, 1526.8) Kv,b1 (kN/
mm) Eq. (46)a 3.64a 3.64a
Kv,b2 9.05a 9.05a
Kax,con,1 (kN/
mm) Fig. 4 484 381
Kax,con,2 600 412
Kθ,c (per plane
of rods) (kNm/
rad) Eq. (17) 6291 8825
Kθ,b (per plane
of rods) Eq. (33) 9156 9156
Kθ,con (per plane
of rods) Eq. (36) 54,249 40,084
Kθ,tot (per plane
of rods) Eq. (37) 3489 4041
Kθ,tot (entire
connection) (kNm/
rad) Eq. (37): n∙Kθ,tot 6978 8082 Experimental [13] 7120b 6579b
(7541c) Deviation between analytical predictions and test
results − 2.0% 22.8 %
(7.2% c) Input for calculations: ρm=430 kg/m3, d=22 mm; d1=16.1 mm, dnet=0.9∙ 20=18 mm, n=2. Es=210000 N/mm2, kv≈kv,90=300 N/mm2 [26], Is= 3298 mm4, lch=55.1 mm (Eq. (47)).
a The vertical stiffness was calculated by the following more accurate form of Eq.
(46) which takes into account the fact that the diameter in the free length is dnet
but the core diameter of the embedded rod is d1: Kv =3∙m∙kv∙lch∙(λ0+m)/(λ04+ 4∙λ03∙m+6∙λ02∙m+6∙λ0∙m+3∙m2), m=dnet4/d14
Compared to Eq. (46) with d1, this equation results in approx. 20% higher values of Kv, approx. 3% higher values in Kθ,b and approx. 1.5% higher values in Kθ,tot i.
e. the use of Eq. (46) with d1 would be a good approximation.
b Measurement based on two pairs of inclinometers attached on both sides of the beam and the column, see Fig. 1(d).
c Measurement based on digital image correlation.
Table 4
Tests with rings as coupling parts (Fig. 1 and Fig. 3) – Forces and utilization ratios at failure.
Parameter Reference S35-55–10 S55-70–10
Failure moment (test),Mu
[13] Fig. 3 78.8 kNm 133.3 kNm
Panel zone Fig. 3
Vu,column a 165.6 kN 280.1 kN
τv,u,column =1.5 ∙Vu,column/b∙h 3.94 N/mm2 6.67 N/mm2
Utilization ratios in each rod Axial load in rods (W:
withdrawal / S:steel)
Fax,c1/Fax,R,c1 Eq. (18)1/Eq.
(43) W:26.4%/
S:32.2% W:53.2%/
S:44.8%
Fax,c2/Fax,R,c2 Eq. (18)2/Eq.
(43) W:97.7%/
S:36.5% W:68.7%/
S:48.1%
Fax,c3/Fax,R,c3 Eq. (19)1/Eq.
(43) W:97.7%/
S:36.5% W:68.7%/
S:48.1%
Fax,c4/Fax,R,c4 Eq. (19)2/Eq.
(43) W:26.4%/
S:32.2% W:53.2%/
S:44.8%
Fax,b1/Fax,R,b1 Eq. (34)1/Eq.
(43) W:28.3%/
S:48.6% W:48.0%/
S:82.3%
Fax,b2/Fax,R,b2 Eq. (35)1/Eq.
(43) W:28.3%/
S:48.6% W:48.0%/
S:82.3%
Lateral load in rods
Fv,b1/Fv,R,b1 Eq. (34)2/Eq.
(48) 30.4% 51.4%
Fv,b2/Fv,R,b2 Eq. (35)2 /Eq.
(48) 22.2% 37.5%
Steel rings
Fx=Fx,con (per ring) Eq. (2) 87.5 kN 148.1 kN
a Determined for the moment corresponding to the centroid of the column, i.e.
Mu =Fjack,u∙(2+0.225)m.
Ultimate steel strength [13]: fu,mean=952 N/mm2 →Ftens,R=As∙fu,mean=( π∙ 16.12/4)
∙952∙10−3 kN =193.8 kN.
Mean embedment strength (for the same rods in GL30c) [26]: fh≈fh,90=17.2 N/mm2, def =1.1∙d1.
Mean yielding moment of the rods [26]: My,R=7.63∙105 N ∙ mm.
Note: No buckling failures of the rods subjected to compression were observed in the tests [13]. The existing model for buckling of self-tapping screws [27] has not been verified for threaded rods.
1,2 1st and 2nd part of the equation respectively.
line resulting in two connecting parts, which could be wrapped and clamped around the rods, and fastened together with bolts (Fig. 1(d)).
In order to evaluate the stiffness and the resistance of the rings, simplified 3D Finite Element (abbr. FE) simulations were carried out by use of Abaqus software [24]. FE simulations were carried for the rings used in each test (Ring35-55 for test S35-55-10 and Ring55-70 for test S55-70-10), both for tensile and compressive loading. The FE models for Ring35-55 are given as an example in Fig. 4(a and b). As a simplification, the washers were assumed tied to the ring. Moreover, the ring was modelled as a whole part, i.e. the slicing of the ring about its symmetry line as shown in Fig. 1(d) was disregarded in the FE models. The rings and the washers were meshed with 8-node, linear, steel, brick elements.
The material properties of steel for the FE models are given in Table 2.
The plasticity of steel was taken into account by use of the stress-plastic
strain levels given in Table 2 for S355 [25]. On the column side, the washers were assumed translationally restrained in the nodes that were in contact with the nuts (the nuts were not included in the model). On the beam side, the washers were subjected to monotonic horizontal displacement applied also in the contact nodes with the nuts. The force- displacement curves of the rings according to the FE results are sum- marized in Fig. 4(c). The elastic stiffness values with respect to hori- zontal displacement are Kax,con,1=484 kN/mm (tension) and Kax,con,2= 600 kN/mm (compression) for Ring35-55, and Kax,con,1=381 kN/mm (tension) and Kax,con,2=412 kN/mm (compression) for Ring55-70.
Table 3 presents in detail the analytical predictions of the component method with respect to the rotational stiffness. The analytical pre- dictions are in good agreement with the experimental results; they slightly underestimate the rotational stiffness for test S35-55-10 and they slightly overestimate the rotational stiffness for test S55-70-10 (in this case the stiffness was also measured by use of digital image corre- lation, giving a slightly higher value than the one based on inclinometers).
The analytical predictions for the shear stresses in the column and the utilization ratios in the rods are summarized in Table 4. The failure moments were 78.8 kNm and 133.3 kNm for tests S35-55-10 and S55- 70-10, respectively. Both specimens failed due to fracture in the panel zone of the column (Fig. 3(d)) as a result of combined shear and tensile stresses perpendicular to grain acting in the panel zone. The influence of tensile stresses perpendicular to grain may be identified by the fact that fracture initiated in the upper part of the panel zone in which threaded rods were subjected to tensile loads. On the contrary, crack opening is prevented in the lower part of the panel zone due to the presence of compressive stresses.
The shear forces in the column can be estimated by the simplified shear force diagram in Fig. 3(b). Here, the lever arm to the centroid of the column was used to determine the shear force in the column. In test S35-55-10 the estimated shear stress in the column at failure was 3.94 N/mm2, i.e. lower compared to the mean shear strength of spruce glu- lam [28]. This may be explained by the occurrence of high concentra- tions of tensile stresses perpendicular to grain at the tip of rods c2 and c3
(note that the tips of rods c2 and c3 are very close to the centroid of the column, i.e. at the theoretical position of maximum shear stress). In test S55-70-10, the estimated shear stress in the column at failure was 6.67 N/mm2, i.e. significantly higher than test S35-55-10. This increased strength may be explained by the fact that rods are continuous in this case and they may act as reinforcements [29], especially for rods c1 and c2 which are located in the upper part of the panel zone which was subjected to tension.
Table 4 also presents the estimated utilization ratios of all rods at failure. At the column side, all estimated axial forces in the rods (Eqs.
(18) and (19)) were smaller than the corresponding axial capacities (Eq.
Fig. 5. Details of connection details with steel connector [21]
Table 5
Connection with steel connector (Fig. 5) –Stiffness predictions vs experimental results.
Stiffness per plane of rods Entire connection
Parameter Kθ,c (kNm/
rad) Kθ,b (kNm/rad) Kθ,tot (kNm/rad) n∙Kθ,tot (kNm/rad)
Analytical Predictions a 6353 (Eq.
(17)) 7820 (Eq. (33)) 3358b (Eq. (37)) 6716 (n=2)
Mean Experimental
results c,d [21] 6395 d (5
tests) 8720 (3 reinforced beams)8125 (2
unreinforced beams) 3809 (3 reinforced beams)3060 (2
unreinforced beams)3510(All 5 tests) 7618 (3 reinforced beams)6120(2 unreinforced beams)7020(All 5 tests) aTest parameters, input for calculation [21], confer also Fig. 2(a) and Fig. 5. αc1 =αc4 =55◦, αc2 =αc3 =70◦, lc1=lc4=550 mm, lc2=lc3=480 mm, lb1=lb2= 900 mm. l0,c1=65 mm, l0,c2=50 mm, l0,c3=30 mm, l0,c4=40 mm, l0,b1=45 mm, l0,b2=25 mm. zc=380 mm, zb=405 mm, Lv=2300 mm, Beam and column GL30c [23]: ρm=430 kg/m3, MC≈12%. Threaded rods: the same as in Section 4.1.
b Input for calculation [21]: Kθ,con=160000 kNm/rad (entire connection), based on linear elastic FE analysis.
cRotations obtained by use of displacement transducers placed on the top and the bottom of the connection on both sides.
dBased on all 5 tests:CoV(Kθ,c) =18.0%, CoV(Kθ,b) =9.8%, CoV(Kθ,tot) =13.5%.
